# Chapter 9

## Leo Mariani

Chapter nine introduces us to some very interesting topics in Geometry. It is interesting that during my studies of high school geometry, I do not recall any discussion of Euclid's five general axioms! Though they were alluded to from time to time, there was no formal discussion of origination and purpose like the one described in the text. I am glad that students in the class decided to write an updated version of the classical geometry text because I look forward to reading it on line. I had never heard of the distinction between intrinsic and extrinsic geometry before. Was this idea developed strictly by Gauss or was there prior foundation for this field? I can imagine that Gauss' theorems for relating the intrinsic and extrinsic geometries of surfaces must be almost incomprehensible to all but the super-elite. Doesn't the fact that gravity seems to originate from the distortion of space seem to imply that the space we live in is curved and not flat? On page 191 you refer to a diagram of two perpendicular planes meeting at a point. Where ios this diagram? Can it be proven that the marble hand exhibits no particular handedness in four space? Is the handedness just a matter of perspective? I am amazed that the mobius band was not invented until 1840. No one else had thought of such a simple construction technique until this time? This must be related to the fact that violations of Euclid's rules were considered heresy. I am having great difficulty picturing the merger of the boundaries of a disk and a mobius strip. I would like to see a step by step construction of the real projective plane by a graphics computer. This is an excellent concluding chapter as there are many easy to understand and clear concepts brought to life from the previous eight chapters.